# Lecture - Linear Algebra I MT22, XIV

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/lectures/14/ · Updated: 2022-12-06 · Tags: uni, lecture

### Flashcards
What is the Cauchy-Schwartz inequality relating $v$ and $w$ in an inner product space?::
$$
|\langle v, w \rangle| \le ||v|| \text{ } ||w||
$$

What is $||\alpha \underline{v}||$ equal to?::
$$
|\alpha|\text{ }||v||
$$

How does the triangle inequality generalise for a norm $||u + v||$?::
$$
||u + v|| \le ||u|| + ||v||
$$

What is the statement of the inequality of arithmetic and geometric means for $x_1, x_2, \ldots, x_n$?::
$$
\frac{x_1 + x_2 + \ldots + x_n}{n} \ge \sqrt[n]{x_1\cdot x_2 \cdot\ldots\cdot x_n}
$$

When does equality hold for the arithmetic and geometric means
$$
\frac{x_1 + x_2 + \ldots + x_n}{n} \ge \sqrt[n]{x_1\cdot x_2 \cdot\ldots\cdot x_n}
$$
?::
When $x_1 = x_2 = \ldots = x_n$.

(Best approximation in inner product spaces) If $V$ is an inner product space and $W$ is a subspace of $V$, if for any $v \in V$ then $w\in W$ satifies $\langle v - w, r \rangle = 0 \forall r \in W$ then what must be true about the distance from $v$ to $w$ and from $v$ to any other vector $u \in W$?::
$$
||v - w|| \le ||v - u||
$$

In a vector space over $\mathbb{C}$, the axioms for an inner product space are different. What is the axiom about linearity?::
$$
\langle \alpha u + \beta v, w\rangle = \alpha\langle u, w\rangle + \beta \langle v, w \rangle
$$

In a vector space over $\mathbb{C}$, the axioms for an inner product space are different. What is the axiom about symmetry?::
$$
\langle u, v \rangle = \overline{\langle v, u \rangle}
$$

Why don’t the axioms for an inner product space over $\mathbb{C}$ specify that the inner product is bilinear?::
Because the symmetric conjugate rule means that it isn’t.

In a vector space over $\mathbb{C}$, the axioms for an inner product space are different. What is the axiom about positive definiteness?::
$$
\langle v, v \rangle \ge 0 \text{ with } v = 0 \iff \langle v, v \rangle = 0
$$

Why is $\langle v, v \rangle$ guarenteed to be real in an complex inner product space?::
Because the symmetric-conjugate condition.

What is the conjugate transpose, $A^*$?::
$$
A^* = \overline{A^\intercal}
$$

What does it mean for a matrix $A$ to be unitary?::
$$
A^\*A = AA^\* = I
$$

What’s the analogue of the dot product $x \cdot y = x^\intercal y$ for a complex vector space?::
$$
x^* y
$$

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